Dragon Notes

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  UNDER CONSTRUCTION

Linear Systems


Waves


Subtopics


Basics

State-space equations

\(\ds\boxed{\begin{align}\dot{\bn{x}}&=\bn{Ax}+\bn{Bu}\\ \bn{y}&=\bn{Cx}+\bn{Du}\end{align}}\)\(\ds\ \ \begin{align}&\t{State equation}\\ &\t{Output equation}\end{align}\)

\(\hspace{70px}\) for \(t\geq t_0\) and IC's \(\bn{x}_0(t)\), where
\(\ds \begin{align} \bn{x} &= \t{state vector} &&& \bn{A} &= \t{system matrix} \\ \dot{\bn{x}} &= \t{time-derivative of state vector} &&& \bn{B} &= \t{input matrix} \\ \bn{y} &= \t{output vector} &&& \bn{C} &= \t{output matrix} \\ \bn{u} &= \t{input or control vector} &&& \bn{D} &= \t{feedforward matrix} \end{align}\)

\(\hspace{70px}\) System matrix: relates how the current state \(\bn{x}\) affects the state change \(\bn{x}'\)
\(\hspace{70px}\) Control matrix: determines how the system input affects the state change
\(\hspace{70px}\) Output matrix: relates the system state to the system output
\(\hspace{70px}\) Feedforward matrix: determines the direct relationship between the system input and output. For feedback systems, \(\bn{D}=0\)
\(\hspace{70px}\) State transition matrix: \(e^{\bn{A}t}\) - matrix whose product with the state vector \(\bn{x}\) gives at an initial time \(t_0\) gives \(\bn{x}\) at a later time \(t\)

\(\hspace{70px}\) Can be computed via: (1) diagonal matrices: raise each diagonal entry of the (diagonal) matrix \(\bn{A}\) as a power of \(e\);
\(\hspace{230px}\) (2) inverse Laplace: \(e^{\bn{A}t}=\mathcal{L}^{-1}[(s\bn{I}-\bn{A})^{-1}]\)

For a second-order LTI system with a single input \(v(t)\), the state equations could take on the following form:

\(\ds\begin{align}\frac{dx_1}{dt}&=a_{11}x_1+a_{12}x_2+b_1v(t),\\ \frac{dx_2}{dt}&=a_{21}x_1+a_{22}x_2+b_2v(t),\end{align}\)
where \(x_1\) & \(x_2\) are the state variables. If there is a single output, the output equation could take on the following form:

\(\ds y=c_1x_1+c_2x_2+d_1v(t)\)
State variables are non-unique, must be linearly-independent, and chosen in some minimum number
\(\ds \boxed{\begin{align} \dot{x} &= ax+by \\ \dot{y} &= cx+dy \end{align}}\)
\(\dsup \boxed{\bn{\dot{x}}=\bn{Ax}}\)

State space \(\rightarrow\) transfer function
[𝕯

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\(\ds\boxed{\begin{matrix}\bn{\dot{x}}=\bn{Ax}+\bn{Bu}\\\bn{y}=\bn{Cx}+\bn{Du}\end{matrix}\Rightarrow \begin{matrix}\begin{align}\bn{X}(s)&=(s\bn{I}-\bn{A})^{-1}\bn{BU}(s)\\ \bn{Y}(s)&=[\bn{C}(s\bn{I}-\bn{A})^{-1}\bn{B}+\bn{D}]\bn{U}(s)\end{align}\end{matrix}}\)
\(\ds\boxed{T(s)=\frac{Y(s)}{U(s)}=\bn{C}(s\bn{I}-\bn{A})^{-1}\bn{B}+\bn{D}}\)

  - \(\t{I.C.}\)'s \(=0\)
  - \(\bn{U}(s)\) & \(\bn{Y}(s)\) are scalar functions \(U(s)\) & \(Y(s)\)
State-space representation

A state-space representation can be obtained as follows:
\(\bb{1}\) Select a subset of all possible system variables, call them state variables
\(\bb{2}\vplup\) For an \(n\)-th order system, write \(n\) simultaneous, first-order differential equations in terms of state variables
\(\bb{3}\vplup\) If IC's of all the state variables at \(t_0\) are known, as well as the system input for \(t\geq t_0\), the diff-eq's can be solved for the state variables for \(t\geq t_0\)
\(\bb{4}\vplup\) Algebraically combine the state variables with the input and find all the other system variables for \(t\geq t_0\), call it the output equation
\(\bb{5}\vplup\) The state equations and output equations form the state-space representation of the system

Controllability

Ability to go from any present state to any future state:
\(\phi(x(t),u[t,T])=c\)
A system is controllable iff for any present state \(x(t)\), there exists an input \(u[t,T]\) which takes the system to any desired state \(x(T)=c\).
Controllability Conditions

Grammian condition: (time-variant & time-invariant)
\(\ds \bn{G}_c = \int_t^T \bn{\Phi}(\bn{T},\tau)\bn{B}(\tau)\bn{B'}(\tau)\bn{\Phi'}(\bn{T},\tau)d\tau\) is nonsingular \((\t{det}\neq 0)\)
Test matrix\(\Up\):
asm
time-invariant sys.
\(\Up \bn{T}_c = [\bn{B},\bn{AB},...,\bn{A}^{(n-1)}\bn{B}]\) is of rank \(n\) \((n\) lin-indep. cols/rows\()\)

Observability

Ability to determine the initial state from the output:
A system is observable iff for any \(x(t)\) there is an operation on \(y[t,T]\) by which one can determine \(x(t)\)
Observability Conditions

One-to-one: (time-variant & time-invariant)
If two or more initial states yield the same output, the system's not observable
Grammian condition\(\Up\): (time-invariant & possibly time-varying):
\(\ds \bn{G}_o = \int_{t}^{T}\bn{\Phi'}(\bn{T},\tau)\bn{C'}(\tau)\bn{C}(\tau)\bn{\Phi}(\bn{T},\tau)d\tau\) is nonsingular \((\t{det}\neq 0)\)
Test matrix\(\Up\):
asm
time-invariant sys.
\(\Up \bn{T}_o = [\bn{C'},\bn{A'C'},...,\bn{A'}^{(n-1)}\bn{C'}]\) is of rank \(n\) \((n\) lin-indep. cols/rows\()\)

Fourier Transform Visual
(\(\mathcal{F}[\delta (t)] = 1\))
Fourier transform visual
Fourier transform visual
Fourier transform visual






Dragon Notes,   Est. 2018     About

By OverLordGoldDragon